In this section, we review results concerning the information theoretic capacity of time-varying wireless channels, with channel side information available at both the transmitter and receiver. We consider two scenarios, i.e., for single- user systems and for multiple-access systems. Note that as the information theoretic capacity is of concern, the data arrival statistics and buffer condition are not taken into consideration.
2.2.1 Single-user Scenario
Consider a single-user system in which a transmitter sends data to a receiver over a time-varying wireless channel. As before, the instantaneous channel power gain is denoted byγ. The probability distribution of γ ispΓ(γ).
Assuming that the instantaneous value ofγis available at both the transmit- ter and the receiver, a power control policy is defined as a functionP(γ) which
26 sets the transmit power when the channel gain is γ. Suppose that all power control policies employed must satisfy the average transmit power constraint
Z
γ
P(γ)pΓ(γ)dγ ≤ P . (2.7)
We are interested in the capacity of this system, i.e., the maximum transmission rate that can be achieved with some power control and coding schemes such that the probability of error is arbitrarily small. In [GV97], Goldsmith and Varaiya give the following definition for the fading channel capacity and subsequently prove a channel coding theorem and converse.
Definition 2.2.1. ([GV97]) Given the average power constraint (2.7), define the time-varying channel capacity by
C(P) = max
P(γ):
R
γP(γ)pΓ(γ)dγ ≤ P
Z
γ
Wlog2
1 + P(γ)γ NoW
pΓ(γ)dγ. (2.8) In particular, it is shown in [GV97] that the power control policy which maxi- mizes (2.8) exhibits the following interesting structure.
P(γ) NoW =
1
γ∗ − γ1, γ ≥γ∗ 0, γ < γ∗.
(2.9) Equation (2.9) tells us that there is a cutoff valueγ∗below which no transmission should be carried out. Above this cutoff value, the power allocation follows a water-filling ([Gal68, BV04]) structure in time, with more transmit power (and rate) being allocated when the channel gain increases. The value ofγ∗ depends on the channel gain distribution and the power constraint through
Z ∞
γ∗
NoW 1
γ∗ − 1 γ
pΓ(γ)dγ = P . (2.10)
Substituting (2.9) into (2.8) gives us the capacity of fading channel, with channel side information at both the transmitter and the receiver. The cod- ing/decoding scheme which achieves this capacity is described in [GV97]. The
main idea is to multiplex multiple coding and modulation schemes, each opti- mized for a particular fade level. The resultant coding/decoding scheme is both variable-power and variable-rate.
We note that the results discussed above are for the information theoretic capacity, which can not be achieved with practical coding/decoding schemes.
In [GC97], Goldsmith and Chua consider a similar problem of communicating over a time-varying channel, but in a practical setup. Adaptive transmission is based on a variable-power variable-rate M-ary quadrature modulation (MQAM) scheme. In particular, by fixing the symbol rate while varying the signal con- stellation size, different transmission rates can be achieved. Similar to the in- formation theoretic setup, the transmission rate and power are varied based on instantaneous channel gain. The objective is to find an adaptive MQAM scheme that maximizes the average transmission rate, subject to the constraints on av- erage transmit power and bit error rate. It is interesting to see that the optimal adaptive MQAM scheme that maximizes the expect transmission rate also fol- lows the water-filling structure [GC97]. In particular, more transmit power (and rate) is allocated when the channel gain increases.
2.2.2 Multiple-access Scenario
The capacity of a multiple-access system is characterized by its capacity region, i.e., the set of all possible rate vectors that can be supported by the system with arbitrarily small probability of error. Within this capacity region, an important performance metric is the sum-of-rate capacity, i.e., the maximum total achievable rates for all users.
Letγ be the vector of instantaneous channel gain ofN users, withγn being
28 the channel gain for user n, n ∈ {1,2, . . . N}. Again, we assume that the instantaneous value ofγ is available at the transmitters and receiver. LetPn(γ) be a power control scheme that set the transmit power for user n when the instantaneous channel gain is γ. In [KH95], Knopp and Humblet study the problem of maximizing the sum-of-rate capacity of a multiple-access system, subject to the average transmit power constraint of each node. In particular, the objective is to maximize
C(P) = Z
γ1
Z
γ2
. . . Z
γN
Wlog2 1 + Xn
n=1
Pn(γ)γn NoW
!
p(γ)dγ, (2.11) subject to
Z
γ1
Z
γ2
. . . Z
γN
Pn(γ)p(γ)dγ ≤ Pn, ∀n ∈ {1,2, . . . N}. (2.12) Note that in (2.11) and (2.12), P = (P1, P2, . . . PN) where Pn is the average power constraint of user n.
It is shown in [KH95] that the power allocation scheme that maximizes (2.11) has the following form.
Pn(γ) NoW =
1
γn∗ − γ1n, γn ≥γn∗, γγn∗n > γγm∗m, m6=n 0, otherwise.
(2.13)
From 2.13, it can be seen that at each time instance, the channel is allocated to at most one user. The user who is assigned the channel must have the relatively best channel gain. Moreover, for the selected user, transmit power is again allocated according to a water-filling structure in time.